A bounded linear operator $T:X\to {\ell }_{\infty }(\Gamma )$ is exactly a uniformly bounded family of bounded linear functionals $f_{\gamma }\in X^{\ast }$ through $(Tx)(\gamma )=f_{\gamma }(x)$. Evaluation shows $\Vert f_{\gamma }\Vert \le \Vert T\Vert$; conversely the family bound gives $\Vert Tx\Vert \le (\sup \Vert f_{\gamma }\Vert )\Vert x\Vert$, proving equality. Taking all functionals in the dual unit ball gives an isometric embedding by the Hahn-Banach theorem.

The bounded piecewise-linear shear layer has exponentially growing modes exactly when $h<1/2$. Its squared phase speed tends to $2h-1$ as $kL\to 0$ and to $1$ as $kL\to \infty$. For $h<1/2$, sufficiently long nonzero waves are therefore unstable. Conversely $J-XY>0$, and for $h\ge 1/2$, $q\ge \alpha (1-h)>Y$, so $H-Y>0$ for every positive wavenumber and $c^2>0$. This proves the threshold without assuming monotonicity of the dispersion curve.

This construction adapts compactly supported orthonormal wavelets to an interval by finite changes near its endpoints. It preserves local support, polynomial reproduction, nested approximation spaces and orthogonal detail spaces. The resulting basis contains coarse scaling functions, unchanged interior wavelets, and finitely many boundary wavelets per endpoint and per scale, without forcing periodic or zero boundary values.

The Born series is the Neumann series expansion of the Lippmann-Schwinger equation. Its terms describe successive orders of wave scattering; convergence is ensured, for example, by a scattering operator norm smaller than one.

A finite positive Borel measure on a compact Hausdorff space is regular when its values on Borel sets can be approximated from inside by compact sets and from outside by open sets. For signed or complex measures require this property of the variation measure. These are the measures representing the continuous dual space of $C(K)$ in the Riesz-Markov-Kakutani representation theorem.

A Borel probability measure is a Borel measure with total mass one. On a compact metric space it is determined by its integrals against continuous functions. On that compact metric space, the family of Borel probability measures is compact for weak convergence of probability measures.

On a closed Riemannian manifold with nonnegative Ricci curvature, integration of the Bochner-Weitzenbock formula for one-forms for a harmonic one-form givesBoth terms are nonnegative, so $\nabla \alpha =0$. Riemannian volume density makes the argument valid even without orientation.

With $s=(12)$, the idempotents $e_{\pm }=(1\pm s)/2$ split the right regular module into two three-dimensional projective modules. Their tops are respectively the trivial representation and sign representation. The nilpotent ideal $J(k{S}_3)$ ensures that any nonzero direct summand has nonzero top, so the one-dimensional tops make these projective modules indecomposable. Their successive radical series of a module factors are trivial, sign, trivial and sign, trivial, sign. This decomposition concerns right modules; the block of S3 in characteristic three remains a single algebra.

In a probability measure-preserving system, $f\in L^1$ implies $f(T^{n}x)/n\to 0$ almost everywhere. For every $\epsilon >0$, the tail integral formula for moments gives ${\sum }_{n\ge 1}\mu (|f|>\epsilon n)\le \Vert f{\Vert }_1/\epsilon$. Invariance and the Borel-Cantelli lemma finish the proof. Consequently division by $n^a$ also gives zero for every $a\ge 1$.

Suppose $f_k\to f$ almost everywhere and ${\sup }_k|f_k|\in L^1$ in a probability measure-preserving system. Thenalmost everywhere and in $L^1$, where $\mathcal{I}$ is the invariant sigma-algebra. For the almost-everywhere assertion, bound the terms with index at least $M$ by ${\sup }_{k\ge M}|f_k-f|$, apply the Birkhoff ergodic theorem, and then let $M\to \infty$. The finitely many remaining end terms vanish by the linear growth bound for integrable observables.

For a measure-preserving transformation and real integrable $f$, let $S_{k}f={\sum }_{j<k}f\circ T^j$. For $E_N=\{{\max }_{1\le k\le N}{S}_{k}f>0\}$ the lemma asserts ${\int }_{E_N}fd\mu \ge 0$. Passing to the increasing union gives the corresponding infinite-supremum inequality.

Suppose a semi-additive category has countable products and coproducts and the canonical comparison between them is always invertible. For countably many copies of $A$, let $q:A\to {\coprod }_{i}A$ correspond under that comparison to the diagonal into the product, and let $\sigma :{\coprod }_{i}A\to A$ be the fold map. If $s$ shifts the summands by one, coordinatewise comparison proves $q={\nu }_1+sq$ and $\sigma s=\sigma$. Thus $z=\sigma q$ satisfies $z=1_A+z$. In an additive category, cancellation gives $1_A=0$, forcing every hom-set to contain only its zero morphism. The category is equivalent to the one-object, one-arrow category.

In a pointed category with finite product in a category and coproducts in a category and invertible canonical maps $c:A+B\to A\times B$, the unique commutative-monoid enrichment is $f+g=[1_B,1_B]c_{B,B}^{-1}⟨f,g⟩$. Associativity and commutativity follow by comparing fold maps on triple and swapped coproduct injections. Composition distributes by the universal properties. For uniqueness, bilinearity forces $i_1{p}_1+i_2{p}_2=1$ on each biproduct, and composing with the fold and the paired morphism forces the addition formula.

For linear gas drag with aerodynamic stopping times $t_1,t_2$, masses $m_1,m_2$, total mass $m=m_1+m_2$, and relative velocity $\mathbf{v}$, defineIf $\mathbf{w}$ is the gas velocity relative to the centre of mass, the relative drag acceleration is $-\kappa \mathbf{v}+\Delta \mathbf{w}$. The first term dissipates internal specific orbital energy; the second is a differential headwind that vanishes for equal aerodynamic stopping times.

In Ostwald ripening, the higher interfacial chemical potential of small droplets drives diffusion toward larger droplets, increasing the characteristic domain size. The Gibbs--Thomson relation supplies the curvature-dependent surface value, while conserved order-parameter dynamics controls transport through the surrounding phase.

If $D$ is a big divisor and $F$ an effective Cartier divisor, infinitely many $m$ have $H^0(X,mD-F)\ne 0$. The divisor restriction exact sequence bounds the dimension lost upon restriction to $F$ by $O(m^{n-1})$, using the polynomial bound for sections of a fixed divisor; this cannot exhaust the $c{m}^n$ sections along the infinite growth sequence.

A Cartier divisor $D$ on an integral projective variety is big exactly when, for every ample divisor $A$, some positive integer $j$ satisfies $jD\sim A+E$ with $E$ effective. To prove the forward direction, subtract an effective high multiple of $A$ using the section subtraction lemma for big divisors, then add an effective representative of the remaining multiple. For the reverse direction, multiply sections of $qA$ by the section of $qE$ and use the ample Hilbert polynomial.

A Cartier divisor on an integral projective variety is big if and only if some multiple has a complete linear system of a divisor giving a rational map birational onto its image. Kodaira's lemma embeds a very ample subsystem in a suitable multiple. Conversely, $n$ algebraically independent elements among the section ratios give $\left(\genfrac{}{}{0ex}{}{q+n}{n}\right)$ independent degree-$q$ section monomials, where $n=\dim X$. Smoothness is not necessary.

A big real divisor is an actual positive real combination of big Cartier divisors. Equivalently it is real linearly equivalent, or numerically equivalent, to an ample real divisor plus an effective real divisor. The rational approximation of an ample-plus-effective real divisor and Kodaira's lemma connect these formulations. Fujino's notes on big real divisors give the definition also for nonnormal varieties; bigness under finite normalization relates it to section growth.

Let $\nu :X^{\nu }\to X$ be the finite normalization of an integral projective variety. The coherent sheaf $\mathcal{Q}={\nu }_{\ast }{\mathcal{O}}_{X^{\nu }}/{\mathcal{O}}_X$ is supported in dimension at most $n-1$. For a Cartier divisor $D$, the projection formula for sheaves and the resulting long exact sequence in sheaf cohomology giveThe last step uses the polynomial bound for sections of a fixed divisor. Therefore the leading order $m^n$ growth, and hence bigness, is preserved in both directions. This handles nonnormal varieties without assuming a resolution of singularities in positive characteristic.